TT #55 · Parshas Tazria · Aliyah 1 · Passuk 8
English: If, however, her means do not suffice for a sheep, she shall take two turtledoves or two pigeons, one for a burnt offering and the other for a purgation offering. The priest shall make expiation on her behalf, and she shall be pure.
Chabad.org — Chumash, English translation + Rashi for Vayikra 12:8
Torah Temimah cites Yoma 41a for when a pair of bird offerings becomes designated as the burnt offering and the sin offering.
E-Daf — Yoma 41a · Shas.org · Sefaria — Yoma 41a, source text and translation
English: Rav Chisda says that the pair of bird offerings is designated only at one of two points: when the owner takes the birds or when the priest performs the service.
Legal designation
The general rule is that the pair’s two offering roles become fixed only through the owner’s taking or the priest’s service; the next unit supplies the scriptural basis.